Abstract
We study the theory of a Hilbert space H as a module for a unital C*-algebra \({\mathcal{A}}\) from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of \({\mathcal{A}}\). Finally, we show that there is an homeomorphism between the space of types of norm less than 1 in this model companion, and the space of quasistates of the C*-algebra \({\mathcal{A}}\).
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References
Argoty, C.: Forkink and Stability in the Representations of a C*-algebra. (2012, Submitted)
Argoty, C., Ben Yaacov, I.: Model Theory Hilbert spaces Expanded with a Bounded Normal Operator. Avaliable in http://www.maths.manchester.ac.uk/logic/mathlogaps/preprints/hilbertnormal1.pdf (2008)
Argoty C., Berenstein A.: Hilbert spaces expanded with an unitary operator. Math. Log. Quar. 55(1), 37–50 (2009)
Ben Yaacov I.: On perturbations of continuous structures. J. Log. Anal. 1(7), 1–18 (2009)
Ben Yaacov I.: Topometric spaces and perturbations of metric structures. Log. Anal. 1(3-4), 235–272 (2008)
Ben Yaacov I.: Modular functionals and perturbations of Nakano spaces. J. Log. Anal. 1(1), 1–42 (2009)
Berenstein, A., Henson, W.: Model Theory of Probability Spaces with an Automorphism. Submitted, available at http://xxx.lanl.gov/abs/math.LO/0405360 (2004)
Ben Yaacov, I., Berenstein, A., Henson, C.W., Usvyatsov, A.: Model theory for metric structures. model theory with applications to algebra and analysis, volume 2. In: Chatzidakis, Z., Macpherson, D., Pillay, A., Wilkie, A. (eds.) London Math Society Lecture Note Series, vol. 350, pp. 315-427 (2008)
Ben Yaacov, I., Usvyatsov, A.: Continuous first order logic and local stability. Trans. Am. Math. Soc. (2010, to appear)
Ben Yaacov, I., Usvyatsov, A., Zadka, M.: Generic Automorphism of a Hilbert Space. (2004, preprint)
Connes A.: Noncommutative Geometry. Academic Press, New York (1994)
Davidson, K.: C*-algebras by Example. Fields Institute Monographs. American Mathematical Society (1996)
Farah, I., Hart, B., Sherman, D.: Model Theory of Operator Alebras I: Stability. (2011 Preprint) arXiv:0908.2790v1
Farah, I., Hart, B., Sherman, D.: Model Theory of Operator Alebras II: Stability. (2011 Preprint) arXiv:1004.0741v1
Henson, C.W., Tellez, H.: Algebraic Closure in Continuous Logic. Revista Colombiana de Matemáticas Vol 41 [Especial], pp. 279–285 (2007)
Hirvonen, ρ A: Metric Abstract Elementary Classes with Perturbations
Iovino, J.: Stable Theories in Functional Analysis, Ph.D. Thesis. University of Illinois (1994)
Pedersen, G.: C*-algebras and their Automorphism Groups (1979)
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The author is very thankful to Alexander Berenstein for his help in reading and correcting this work. In the same way, the author wants to thank Itaï Ben Yaacov for discussing ideas in particular cases. Similarly, the author wants to thank C. Ward Henson for his advice and help, and for sharing with the author the ideas underlying Sect. 2.22. Finally, the author wants to thank Andrés Villaveces for his interest and for sharing ideas on how to extend the results of this paper.
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Argoty, C. The model theory of modules of a C*-algebra. Arch. Math. Logic 52, 525–541 (2013). https://doi.org/10.1007/s00153-013-0330-2
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DOI: https://doi.org/10.1007/s00153-013-0330-2